Question

Calculate the A.M. and S.D. for the following distribution:

Class:	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency:	18	16	15	12	10	5	2	1

Answer 1

Class	\[f_i\]	Midpoint \[\left( x_i \right)\]	\[u_i = \frac{x_i - 35}{10}\]	\[f_i u_i\]	\[f_i {u_i}^2\]
0−10	18	5	-3	- 54	162
10−20	16	15	-2	- 32	64
20−30	15	25	-1	- 15	15
30−40	12	35	0	0	0
40−50	10	45	1	10	10
50−60	5	55	2	10	20
60−70	2	65	3	6	18
70−80	1	75	4	4	16
	\[\sum f_i = 79\]			\[\sum f_i u_i = - 71\]	\[\sum f_i {u_i}^2 = 305\]

\[\bar{X} = a + h\left( \frac{\sum f_i u_i}{N} \right) = 35 + 10\left( \frac{- 71}{79} \right) = 26 . 01\]

AM = 26.01

\[\sigma^2 = h^2 \left[ \frac{\sum f_i {u_i}^2}{N} - \left( \frac{\sum f_i u_i}{N} \right)^2 \right] = 100\left[ \frac{305}{79} - \frac{5041}{6241} \right] = 305 . 20\]

\[\sigma = \sqrt{305 . 20} = 17 . 47\]